In this section, we prove wasserstein-stability.
The section is structured as follows.
In signed-wasserstein-distance-subsection we define the signed 1-Wasserstein distance \(\widehat{d_W}\)  and establish some of its main properties.
In presentation-distance-subsection we recall the definition of the 1-presentation distance \({d_I^1}\)  introduced by Bjerkevik and Lesnick in [1]}.
In proof-of-wasserstein-stability-subsection we recall necessary background from [1]} and prove wasserstein-stability.
The key results from [1]} that we use are recalled below as useful-universality-lp and grid-lemma.
